Section 4.7 Piecewise Functions 217 Piecewise Functions 4.7 Writing Equations for a Function Work with a partner. a. Does the graph represent y as a function of x? Justify your conclusion. b. What is the value of the function when x = 0? How can you tell? c. Write an equation that represents the values of the function when x ≤ 0. f (x) = , if x ≤ 0 d. Write an equation that represents the values of the function when x > 0. f (x) = , if x > 0 e. Combine the results of parts (c) and (d) to write a single description of the function. f (x) = , if x ≤ 0 , if x > 0 Essential Question Essential Question How can you describe a function that is represented by more than one equation? CONSTRUCTING VIABLE ARGUMENTS To be proficient in math, you need to justify your conclusions and communicate them to others. Writing Equations for a Function Work with a partner. a. Does the graph represent y as a function of x? Justify your conclusion. b. Describe the values of the function for the following intervals. , if −6 ≤ x < −3 f (x) = , if −3 ≤ x < 0 , if 0 ≤ x < 3 , if 3 ≤ x < 6 Communicate Your Answer Communicate Your Answer 3. How can you describe a function that is represented by more than one equation? 4. Use two equations to describe the function represented by the graph. x y 2 4 6 6 4 2 −2 −4 −6 −2 −4 −6 x y 2 4 6 6 4 2 −2 −4 −6 −2 −4 −6 x y 2 4 6 6 4 2 −2 −4 −6 −2 −4 −6
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Piecewise Functions TSHS/Chapter 1/L1.3/Day...Section 4.7 Piecewise Functions 221 Writing Absolute Value Functions The absolute value function f(x) = ∣ x ∣ can be written as a
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Section 4.7 Piecewise Functions 217
Piecewise Functions4.7
Writing Equations for a Function
Work with a partner.
a. Does the graph represent y as a function
of x? Justify your conclusion.
b. What is the value of the function when
x = 0? How can you tell?
c. Write an equation that represents the values
of the function when x ≤ 0.
f(x) = , if x ≤ 0
d. Write an equation that represents the values
of the function when x > 0.
f(x) = , if x > 0
e. Combine the results of parts (c) and (d) to write a single description of the function.
f(x) = , if x ≤ 0
, if x > 0
Essential QuestionEssential Question How can you describe a function that is
represented by more than one equation?
CONSTRUCTING VIABLE ARGUMENTSTo be profi cient in math, you need to justify your conclusions and communicate them to others.
Writing Equations for a Function
Work with a partner.
a. Does the graph represent y as a function
of x? Justify your conclusion.
b. Describe the values of the function for the
following intervals.
, if −6 ≤ x < −3
f(x) = , if −3 ≤ x < 0
, if 0 ≤ x < 3
, if 3 ≤ x < 6
Communicate Your AnswerCommunicate Your Answer 3. How can you describe a function
Writing Absolute Value FunctionsThe absolute value function f(x) = ∣ x ∣ can be written as a piecewise function.
f(x) = { −x,
x,
if x < 0
if x ≥ 0
Similarly, the vertex form of an absolute value function g(x) = a ∣ x − h ∣ + k can be
written as a piecewise function.
g(x) = { a[−(x − h)] + k,
a(x − h) + k,
if x − h < 0
if x − h ≥ 0
Writing an Absolute Value Function
In holography, light from a laser beam is
split into two beams, a reference beam and
an object beam. Light from the object beam
refl ects off an object and is recombined
with the reference beam to form images
on fi lm that can be used to create
three-dimensional images.
a. Write an absolute value function that
represents the path of the reference beam.
b. Write the function in part (a) as a
piecewise function.
SOLUTION
a. The vertex of the path of the reference beam is (5, 8). So, the function has the
form g(x) = a ∣ x − 5 ∣ + 8. Substitute the coordinates of the point (0, 0) into
the equation and solve for a.
g(x) = a ∣ x − 5 ∣ + 8 Vertex form of the function
0 = a ∣ 0 − 5 ∣ + 8 Substitute 0 for x and 0 for g(x).
−1.6 = a Solve for a.
So, the function g(x) = −1.6 ∣ x − 5 ∣ + 8 represents the path of the
reference beam.
b. Write g(x) = −1.6 ∣ x − 5 ∣ + 8 as a piecewise function.
g(x) = { −1.6[−(x − 5)] + 8,
−1.6(x − 5) + 8,
if x − 5 < 0
if x − 5 ≥ 0
Simplify each expression and solve the inequalities.
So, a piecewise function for g(x) = −1.6 ∣ x − 5 ∣ + 8 is
g(x) = { 1.6x,
−1.6x + 16,
if x < 5
if x ≥ 5 .
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12. WHAT IF? The reference beam originates at (3, 0) and refl ects off a mirror
at (5, 4).
a. Write an absolute value function that represents the path of the
reference beam.
b. Write the function in part (a) as a piecewise function.
STUDY TIPRecall that the graph of an absolute value function is symmetric about the line x = h. So, it makes sense that the piecewise defi nition “splits” the function at x = 5.
REMEMBERThe vertex form of an absolute value function is g(x) = a ∣ x − h ∣ + k, where a ≠ 0. The vertex of the graph of g is (h, k).